Chemistry Reference
In-Depth Information
the interphase,
P
i
m
i
dn
i
, divided by the interfacial area
A
S
. Substitution of (
11
)
into the total derivative of (
12
) results in:
!
=
X
V
S
dP
A
S
d g ¼
SdT
þ
n
i
d m
i
(13)
i
which is a modified Gibbs Duhem equation for the interphase.
The quantity of the components adsorbed at the interphase is a significant
parameter, whereas the relationship between the extent of adsorption and the
interfacial tension is particularly of interest; this is studied in terms of the Gibbs
adsorption isotherm. At constant temperature and pressure, the Gibbs Duhem
relationship for an interphase is:
X
X
i
G
i
d m
i
A
S
d g ¼
n
i
d m
i
=
¼
(14)
i
n
i
/
A
S
is the quantity of the i-th constituent contained per unit area of the
interphase. Equation (
14
) indicates that spatial partitioning of constituents occurs at
an interface (i.e., one constituent adsorbs preferentially at the interface) and that the
extent of this adsorption is a function of the interfacial tension. The definition of
G
i
,
however, is not exact because it depends on the concentration gradients present
within the interphase, and its magnitude depends on the choice of the dividing
boundary, often referred to as the Gibbs dividing surface.
For a two component system, the Gibbs adsorption isotherm is written as:
where
G
i
¼
d g ¼ G
1
d m
1
þ G
2
d m
2
(15)
Although recognizing that the interfacial region is best considered as an interphase,
the alternative mathematical model is to consider the interface as a plane of
infinitesimal thickness situated between AA
0
and BB
0
of Fig.
12
. This dividing
surface can be considered to be positioned so as to give rise to a simplification of
(
15
). Gibbs [
183
] defined the position of the dividing surface such that the surface
excess of constituent 1 is zero, and hence:
d g ¼ G
0
2
d m
2
(16)
G
0
2
is the surface excess of constituent 2 with the dividing surface so defined.
The equation relates the reduction in interfacial tension directly to the enrichment
of one component within the interphase.
Although the thermodynamic description of an interphase is an invaluable tool,
it is rarely used. The traditional approach of Gibbs requires the use of a dividing
surface to which interfacial properties are referenced. This method is burdened with
notational and conceptual difficulties [
184
]. As alternative but equivalent method of
treating interphase thermodynamics was developed by Cahn [
185
], which avoided
where
